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Exponential Functions and Models. Lesson 5.3. Contrast. Definition. An exponential function Note the variable is in the exponent The base is a C is the coefficient, also considered the initial value (when x = 0). Explore Exponentials.

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Definition l.jpg
Definition

  • An exponential function

    • Note the variable is in the exponent

    • The base is a

    • C is the coefficient, also considered the initial value (when x = 0)


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Explore Exponentials

  • Given f(1) = 3, for each unit increase in x, the output is multiplied by 1.5

  • Determine the exponential function


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Explore Exponentials

  • Graph these exponentials

  • What do you think the coefficient C and the base a do to the appearance of the graphs?


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Contrast Linear vs. Exponential

  • Suppose you have a choice of two different jobs at graduation

    • Start at $30,000 with a 6% per year increase

    • Start at $40,000 with $1200 per year raise

  • Which should you choose?

    • One is linear growth

    • One is exponential growth


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Which Job?

  • How do we get each nextvalue for Option A?

  • When is Option A better?

  • When is Option B better?

  • Rate of increase a constant $1200

  • Rate of increase changing

    • Percent of increase is a constant

    • Ratio of successive years is 1.06


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Example

  • Consider a savings account with compounded yearly income

    • You have $100 in the account

    • You receive 5% annual interest

View completed table


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Compounded Interest

  • Completed table


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Compounded Interest

  • Table of results from calculator

    • Set Y= screen y1(x)=100*1.05^x

    • Choose Table (♦Y)

  • Graph of results


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Assignment A

  • Lesson 5.3A

  • Page 415

  • Exercises 1 – 57 EOO


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Compound Interest

  • Consider an amount A0 of money deposited in an account

    • Pays annual rate of interest r percent

    • Compounded m times per year

    • Stays in the account n years

  • Then the resulting balance An


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Exponential Modeling

  • Population growth often modeled by exponential function

  • Half life of radioactive materials modeled by exponential function


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Growth Factor

  • Recall formulanew balance = old balance + 0.05 * old balance

  • Another way of writing the formulanew balance = 1.05 * old balance

  • Why equivalent?

  • Growth factor: 1 + interest rate as a fraction


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Decreasing Exponentials

  • Consider a medication

    • Patient takes 100 mg

    • Once it is taken, body filters medication out over period of time

    • Suppose it removes 15% of what is present in the blood stream every hour

Fill in the rest of the table

What is the growth factor?


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Decreasing Exponentials

  • Completed chart

  • Graph

Growth Factor = 0.85

Note: when growth factor < 1, exponential is a decreasing function


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Solving Exponential Equations Graphically

  • For our medication example when does the amount of medication amount to less than 5 mg

  • Graph the functionfor 0 < t < 25

  • Use the graph todetermine when


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General Formula

  • All exponential functions have the general format:

  • Where

    • A = initial value

    • B = growth rate

    • t = number of time periods


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Typical Exponential Graphs

  • When B > 1

  • When B < 1


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Using e As the Base

  • We have used y = A * Bt

  • Consider letting B = ek

  • Then by substitution y = A * (ek)t

  • Recall B = (1 + r) (the growth factor)

  • It turns out that


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Continuous Growth

  • The constant k is called the continuous percent growth rate

  • For Q = a bt

    • k can be found by solving ek = b

  • Then Q = a ek*t

  • For positive a

    • if k > 0 then Q is an increasing function

    • if k < 0 then Q is a decreasing function


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Continuous Growth

  • For Q = a ek*t Assume a > 0

  • k > 0

  • k < 0


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Continuous Growth

  • For the functionwhat is thecontinuous growth rate?

  • The growth rate is the coefficient of t

    • Growth rate = 0.4 or 40%

  • Graph the function (predict what it looks like)


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Converting Between Forms

  • Change to the form Q = A*Bt

  • We know B = ek

  • Change to the form Q = A*ek*t

  • We know k = ln B (Why?)


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Continuous Growth Rates

  • May be a better mathematical model for some situations

  • Bacteria growth

  • Decrease of medicine in the bloodstream

  • Population growth of a large group


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Example

  • A population grows from its initial level of 22,000 people and grows at a continuous growth rate of 7.1% per year.

  • What is the formula P(t), the population in year t?

    • P(t) = 22000*e.071t

  • By what percent does the population increase each year (What is the yearly growth rate)?

    • Use b = ek


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Assignment B

  • Lesson 5.3B

  • Page 417

  • Exercises 65 – 85 odd


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